Abstract
We study abelian repetitions in partial words, or sequences that may contain some unknown positions or holes. First, we look at the avoidance of abelian pth powers in infinite partial words, where p > 2, extending recent results regarding the case where p = 2. We investigate, for a given p, the smallest alphabet size needed to construct an infinite partial word with finitely or infinitely many holes that avoids abelian pth powers. We construct in particular an infinite binary partial word with infinitely many holes that avoids 6th powers. Then we show, in a number of cases, that the number of abelian p-free partial words of length n with h holes over a given alphabet grows exponentially as n increases. Finally, we prove that we cannot avoid abelian pth powers under arbitrary insertion of holes in an infinite word.
This material is based upon work supported by the National Science Foundation under Grant No. DMS–0754154. The Department of Defense is also gratefully acknowledged. We thank Dimin Xu for very valuable comments and suggestions. A research assignment from the University of North Carolina at Greensboro for the first author is gratefully acknowledged. Some of this assignment was spent at the LIAFA: Laboratoire d’Informatique Algorithmique: Fondements et Applications of Université Paris 7-Denis Diderot, Paris, France. A World Wide Web server interface has been established at www.uncg.edu/cmp/research/abelianrepetitions2 for automated use of the program.
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Blanchet-Sadri, F., Simmons, S. (2011). Avoiding Abelian Powers in Partial Words. In: Mauri, G., Leporati, A. (eds) Developments in Language Theory. DLT 2011. Lecture Notes in Computer Science, vol 6795. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22321-1_7
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DOI: https://doi.org/10.1007/978-3-642-22321-1_7
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